Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 42 (2001), No. 1, 71-87.
The Multiple Conical Surfaces
Josef Schicho
Research Institute for Symbolic Computation
Johannes Kepler University, A-4040 Linz, Austria
e-mail: Josef.Schicho@risc.uni-linz.ac.at
http://www.risc.uni-linz.ac.at/people/jschicho
Abstract. We give a classification of the surfaces that can be generated by a moving conic in more than one way. It turns out that these surfaces belong to classes
which have been thoroughly studied in other contexts (ruled surfaces, Veronese
surfaces, del Pezzo surfaces).
1. Introduction
Figure 1: A multiple conical surface
A pencil of conics is a one-parameter family of conics. A conical surface is a surface which
is generated as the union of conics in a pencil. A multiple conical surface is a surface which
is a conical surface in more than one way. For instance, the surface in Figure 1 is generated
by two families of circles passing through a fixed point. The horizontal sections form a third
generating family, consisting of hyperbolas (see Example 5 below).
c 2001 Heldermann Verlag
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J. Schicho: The Multiple Conical Surfaces
The class of multiple conical surfaces includes several classical examples, such as the
quadric surfaces, the torus, the cyclides of Dupin (see [14]), the surfaces of Darboux [6], the
double surfaces of Blutel [9], and a class of surfaces that arises in kinematics and has been
investigated in [15, 16]. They also arise in the surface parametrization problem (see [22, 21]);
there, the problem is to locate one generating pencil, and the fact that there is no unique
one is responsible for the difficulty of this location task (for instance, one needs to introduce
algebraic numbers).
Traditionally, the conical surfaces have been investigated using methods of projective duality and projective differential geometry (see [1, 25, 7, 8, 2, 19, 18]). Here, we use intersection
theory for divisors on surfaces (see [24, 28]).
The classification results obtained this way are quite satisfying: any such surface is algebraic of degree at most 8, and they belong to classes of surfaces which have been thoroughly
investigated from a different point of view: ruled surfaces, Veronese surfaces and del Pezzo
surfaces. We list all possibilities for the exact number of pencils of conics and give other
results of geometrical relevance.
The most abundant subclass of multiple conical surfaces is the one consisting of del Pezzo
surfaces. The del Pezzo surfaces of degree 3 have already been classified in [23, 4] (see also
[3]). For del Pezzo surfaces of higher degree, we use the classification in [5].
This work was supported by the Austrian Science Fund (FWF) in the frame of the
“Adjoints” project and of the project SFB F013. I also want to thank M. Husty for a fruitful
discussion which started this investigation, and for drawing my attention to many classical
examples.
2. General results
We consider complex analytic surfaces in 3-dimensional space. Our first result establishes
that all multiple conical surfaces are algebraic. Subsequently, we show that these surfaces
are rational, and that the generic plane section has genus zero or one. Finally, we introduce
the concept of linear normalization which will be used in subsequent sections.
If we assume that S is a closed analytic subvariety of projective space, than S would
automatically be algebraic by Chow’s theorem (see [12]). But the algebraic nature of multiple
conical surfaces can be shown under much weaker assumptions, which include also the affine
case.
Let U be an open subset of complex 3-dimensional projective space (for instance,
3-dimensional affine space). Let S be a closed analytic subvariety of U of dimension 2.
Let Π be the algebraic variety of dimension 8 parametrizing all conics in projective 3-space.
A family of conics on S is a set of conics parametrized by an analytic subvariety of Π, where
the generic conic is irreducible, such that the intersection of each conic with U lies in S. If
the parametrizing variety is a curve, then we speak of a pencil of conics. We say that the
surface is generated by a pencil of conics if it is the union of all intersections of the conics
in the pencil with U . A surface which is generated by a pencil of conics is called a conical
surface. A multiple conical surface is a surface that is generated by more than one pencil of
conics.
A degenerate case is the plane, which has of course infinitely many pencils of conics
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73
generating it. From now on, we assume that S is not the plane.
Theorem 1. Any multiple conical surface is algebraic.
Proof. In the following, we identify subvarieties of Π with their corresponding families of
conics.
Let Γ1 , Γ2 two (maybe transcendental) pencils in Π generating S. The generic conic in
Γ1 intersects the generic conic in Γ2 (maybe outside U ), hence all conics in Γ1 intersect all
conics in Γ2 .
Let Λ be the family of all conics intersecting each conic in Γ2 , maybe outside U . Then
Λ is an algebraic subvariety of Π, because it is the solution set of infinitely many algebraic
conditions. Moreover Λ contains Γ1 .
Suppose, indirectly, that S is not algebraic. Since the union V of all conics in Λ is
an algebraic set containing S, it must be 3-dimensional. Clearly, there exists an algebraic
subvariety of Λ of dimension 2 such that the union is still 3-dimensional and still contains S.
Let ∆ be such a subvariety.
Let p be a generic point on S. Let C0 be a conic in Γ2 passing through p. For each point
q in a neighborhood of p, there is only a finite set of conics in ∆ passing through q, because
otherwise we would have dim(∆) > 2. Especially, this holds if q∈C0 . If we take the union of
all conics in ∆ passing through q, where q ranges over C0 in a small neighborhood of p, then
we get a two-dimensional set. On the other hand, any conic in ∆ intersects C0 , and therefore
the union must be three-dimensional. This is a contradiction.
Theorem 2. Let S be a multiple conical surface. Then S has several algebraic pencils of
conics.
Proof. Let Γ1 , Γ2 two (maybe transcendental) pencils in Π generating S. Let Λ1 be the
algebraic family of all conics intersecting each conic in Γ2 . Similarly, let Λ2 be the algebraic
family of all conics intersecting each conic in Γ1 . We distinguish two cases.
If Λ1 6=Λ2 , then one can choose algebraic pencils Γ01 ⊂Λ1 , Γ02 ⊂Λ2 , each generating S.
If Λ1 = Λ2 , then Λ1 cannot be a pencil because it contains two distinct pencils. Hence
it is a family of dimension at least two, and one can find two distinct algebraic generating
pencils within it.
The results above show that we may restrict our attention to the algebraic situation.
Consequently, we may assume that U = P3 from now on.
We adopt the terminology (see [13]) that a birationally ruled surface is a surface that
has a rational map to a curve, such that the generic fiber has genus zero (and is therefore
birationally equivalent to a line). By a theorem of Enriques (see [13]), the birationally ruled
surfaces can be characterized as the surfaces with Kodaira dimension −∞, i.e. all plurigeni
equal to 0.
Theorem 3. Any multiple conical surface is rational.
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Proof. Let F1 be an algebraic family of conics generating S, with parameter space T1 , an
algebraic curve. Then the algebraic set
S 0 := {(x, t) | x∈C(t)}⊂S×T1
is a birationally ruled surface, therefore all its plurigenera are 0. The second projection is a
rational map from S 0 to S, hence all plurigenera of S are zero (otherwise, the pullback of a
pluricanonical section would give rise to a pluricanonical section on S 0 which is impossible).
Consequently, S is also a birationally ruled surface.
Suppose, indirectly, that S is irrational. Then any rational curve on S is contained in
a fiber of the ruling. Therefore, there is precisely one pencil of rational curves on S, and S
cannot be multiple conical.
Recall that an algebraic family is called linear iff it consists of the inverse images of the
hyperplanes under a rational map to a projective space (see [13]). Especially, a pencil is
linear iff it consists of the fibers of a rational map to the projective line.
In order to apply intersection theory, we consider a desingularization d : S̃→S of the
multiple conical surface S. We consider divisors (i.e. composite curves where the components
are counted with multiplicity) on the desingularization S̃. Two divisors are called linear
equivalent iff they are contained in a common linear family. This relation is transitiv (see
[13]). The set of all divisors linear equivalent to D forms a linear family, which is denoted
by |D| (the complete linear family defined by D). The set of equivalence classes of divisors
modulo linear equivalence form a group, called the class group. The intersection product is
well-defined on classes.
Theorem 4. Let S be a multiple conical surface. Then S has several linear pencils of conics.
Proof. If all pencils of S are linear, then there is nothing to prove.
The pullbacks of any two curves in an arbitrary pencil are numerically equivalent divisors
on S̃, by continuity. On a nonsingular rational surface, numerical equivalence implies linear
equivalence (see [28]). Therefore, the pullback of a nonlinear pencil on S is contained in a
linear family of divisors on S̃ of dimension at least 2. It is clear that such a linear family
contains an infinite number of linear pencils. By mapping them down to S, we get infinitely
many pencils of conics on S.
Theorem 5. Let S be a multiple conical surface. Then the generic plane section of S has
genus zero or one.
Proof. The pullback H of a generic section is a nonsingular curve on S̃, by Bertini’s theorem
(see [12]). Any linear family on S̃ cuts out a linear family on H. The pullback of a linear
pencil on S cuts out a linear family of degree 2 (because the generic plane section intersects
a generic conic in the pencil in two points) and dimension 1. We distinguish two cases.
If one of these linear families on H is incomplete, then its completion must be of dimension 2. But the existence of a complete linear family of dimension 2 and degree 2 (a g22 in
the terminology of the theory of algebraic curves) implies that H has genus zero (see [26]).
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75
Now, suppose that all these linear families are complete (i.e. g21 ’s). We claim that the
linear families are distinct.
Proof of the claim: choose first a generic point p on S, and then a generic plane E
through p. (Note that we do not give away the generic-ness of the plane by this choice!) Let
C be the plane section. Let P1 be the conic in the linear pencil F1 passing through p, and
let P2 be conic in the linear pencil F2 passing through p. Suppose, indirectly, that the two
linear families cut out the same g21 . Then P1 and P2 intersect the plane E in the same point
q. Because the plane was chosen generic, and the coincidence condition is closed, P1 and P2
intersect any plane through p in the same point. This implies P1 = P2 . Now recall that also
the choice of p was generic, hence F1 = F2 . This proves the claim.
Now, it is well known (see [26]) that any curve that has two different g21 ’s has genus one.
Therefore, H has genus one.
For the rest of the paper, let us fix the following notation. The pullback of a generic plane
section is denoted by H. For any family Fi of conics, let Pi be the pullback of a generic conic.
We have already shown that we have Pi ·H = 2 for all i. Similarly, one can show that
2
H = degS.
Let iH : S̃→Pdim|H| be the rational map defining the linear structure of |H|. The image
variety S⊂Pdim|H| is called the linear normalization of S. A surface is called linearly normal
if is isomorphic to its linear normalization.
Algebraically, the graded coordinate ring of the linear normalization can be obtained as
the subalgebra of the graded integral closure of the given coordinate ring which is generated
in degree 1.
The following theorem is well-known. We include the proof because we could not find a
self-contained proof in the literature. Moreover, it gives geometrical insight to the concept
of linear normalization.
Theorem 6. Any surface is a projection from its linear normalization. The projection is
birational, and it preserves the degree of the surface and the degree of any curve not contained
in the singular locus.
Proof. Since the desingularization map d : S̃→S defines the linear structure of a subfamily
of |H|, it follows that iH is regular and d is iH composed with a projection p to P3 . The
inverse of the projection is iH ◦d−1 (this argument actually shows that the inverse is defined
on all nonsingular points of S). The subfamily of |H| does not have based points, hence the
center of projection is disjoint from the surface S.
If C is a point not contained in the singular locus, its pre-image C on S is birationally
equivalent to it because the inverse map is defined almost everywhere. The map p is a
projection from a subspace disjoint from C, hence deg(C) = deg(C). Taking for C a generic
plane section, we obtain deg(S) = deg(S).
Let S be a surface in P3 , and let S be its normalization. By Theorem 6 above, there is a oneto-one correspondence between pencils of conics on S and pencils of conics on S. Therefore,
it makes sense to classify multiple conical surfaces up to linear normalization.
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3. Ruled surfaces
In this section, we recall the definitions of the Veronese surface and of the ruled surfaces. It
is well-known that the surfaces with generic plane section of genus zero are the projections
of ruled surfaces or of the Veronese surface. We give a complete classification of the multiple
conical surfaces of this type.
The Veronese surface V ⊂P5 is given by the parametrization
(1 : s : t : s2 : st : t2 )
(up to a change of projective coordinates in P5 ). The class group is generated by L, the
divisor t = 0. We have H∼2L and L2 = 1. Thus, the degree of V is H 2 = 4.
For any m, n≥0 such that 2m + n≥2, the ruled surface Rn,m ⊂P2m+n+1 is given by the
parametrization (1 : t : . . . : tm+n : s : st : . . . : stm ). The ruling is formed by the lines where
t is constant. The class group is generated by the ruling P and the cross section B : s = 0.
(The ruled surface R0,1 (a nonsingular quadric) has a second ruling, formed by the lines where
s is constant.) We have H = B + (m + n)P , B·P = 1, P 2 = 0, B 2 = −n. Thus, the degree
of Rn,m is H 2 = 2(m + n) − n = 2m + n.
We will use the following well-known theorem.
Theorem 7. Let S be a surface with generic plane section of genus zero. Then dim|H| =
deg(S) + 1, and the linear normalization of S is either a ruled surface Rn,m or the Veronese
surface (up to projective isomorphism).
Proof. Well-known (see [12]).
Theorem 8. Let S be a multiple conical surface with generic plane section of genus zero.
Then we have one of the following cases.
• S is a quadric surface; any pencil of conics is contained in the 3-dimensional linear
family of plane sections.
• S is a cubic surface; the linear normalization of S is the ruled surface R1,1 ; any pencil
of conics is contained in the 2-dimensional linear family corresponding to |B + P |.
• S is a quartic surface; the linear normalization of S is the Veronese surface; any pencil
of conics is contained in the 2-dimensional linear family corresponding to |L|.
Proof. Let S be the linear normalization of S. By Theorem 7, we know that S is either a
ruled surface Rn,m or the Veronese surface. In the second case, the equation Pi ·H = 2 implies
that Pi ∼L. Hence any pencil of conics is contained in the two-dimensional linear family |L|.
Now, suppose that S = Rn,m . Since the class group is generated by B, P , we may
write Pi ∼ai B + bi P . Because Pi is in a moving linear family, we have P ·Pi = ai ≥0 and
B·Pi = bi − nai ≥0. We also have
Pi · H = (ai B + bi P )·(B + (m + n)P ) = bi + ai m = 2,
and the restriction 2m + n≥2 above. Thus, we are left with the following possibilities.
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1. ai = 0, bi = 2: Then Pi ∼2P , and Pi ·P = 0, hence all curves in |Pi | decompose into two
curves in |P |. This contradicts our assumption that the generic conic of the pencil Fi
is irreducible. Hence this case is impossible.
2. ai = 1, bi = 0, m = 2, n = 0: Then S = R0,2 . Since the complete linear family
|Pi | = |B| has dimension 1, the pencil Fi is equal to |B|.
3. ai = 1, bi = 1, m = 1, n = 0: Then Pi ∼H, and H 2 = 2. Hence S is a quadric surface,
and Fi is a pencil of plane sections. In this case, S = S because a quadric surface is
linearly normal.
4. ai = 1, bi = 1, m = 1, n = 1: Then S = R1,1 , and the pencil Fi is contained in |B + P |.
5. ai = 1, bi = 2, m = 0, n = 2: Again, Pi ∼H, and H 2 = 2. Hence we have another
quadric surface and a pencil of plane sections (the difference to case 3 is that here we
have a singular quadric).
6. ai = 2, bi = 0, m = 1, n = 0: Then Pi ∼2B, and Pi ·B = 0, hence we obtain the same
contradiction as in case 1. So this case is impossible.
Since S is a multiple conical surface, case 2 is also impossible because the ruled surface R0,2
has only one pencil of conics (namely |B|).
Figure 2: The ruled surface x2 y − z 2 = 0 and two pencils of conics on it
Example 1. Here is an example of the second case. The ruled cubic surface S with equation
x2 y − z 2 w = 0 has the linear normalization
u2 − yw = ux − zw = uz − xy = 0.
A parametrization is
(x : y : z : w : u) = (s : t2 : st : 1 : t),
hence we have indeed the ruled surface R1,1 . The parametrization of S is obtained by omitting
the last coordinate.
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For two parameters λ1 , λ2 , we get a conic by substituting s = λ1 t+λ2 in the parametrization. Obviously, these conics form a two-dimensional family of conics. Figure 2 shows the
pencil of parabolas s = constant, the ruling t = constant, and some elements of the pencil of
parabolas s + t = constant.
Figure 3: The Steiner surface and two pencils of conics on it
Example 2. Here is an example of the third case. The Steiner surface with equation
x2 y 2 + y 2 z 2 + z 2 x2 + xyzw = 0
has the linear normalization
uz − xy = vx − yz = uv − y 2 = u2 + uv + uw + x2 =
= v 2 + uv + vw + z 2 = uy + vy + wy + xz = 0.
A parametrization is
(x : y : z : w : u : v) = (s : st : t : −s2 − t2 − 1 : s2 : t2 ),
which shows that linear normalization is a Veronese surface.
As in the previous example, we get a pencil of conics by substituting s = λ1 t + λ2 in the
parametrization. Figure 3 shows the two pencils s = constant. and t = constant.
In the proof above, we also recovered Brauner’s classification [2] of the conical surfaces
which are also ruled surfaces (these are the quadric surfaces and the surfaces R0,2 and R1,1 ).
We refer to [2] for a more detailed discussion of these surfaces.
Remark 1. If S is the ruled surface R1,1 or the Veronese surface, then all pencils are contained in a single linear family of dimension 2. Because any two plane algebraic curves
intersect, we have the following interesting corollary of Theorem 8: if S is a multiple conical
surface of degree greater than 2 with generic plane section of genus zero, then any two algebraic pencils have a common conic. It can also be seen in Figure 3, where it is projected to
a segment of the singular line x = z = 0.
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4. Del Pezzo surfaces
We give a complete classification of the multiple conical surfaces with generic plane section
of genus one, using the theory of del Pezzo surfaces.
A del Pezzo surface is a linearly normal rational surface with generic hyperplane section
of genus one. Equivalently, del Pezzo surfaces can be defined as the linearly normal rational
surfaces of degree d in Pd . They have been investigated by numerous authors [10, 11, 27, 5].
Our motivation for studying del Pezzo surfaces is the following fact.
Proposition 1. Let S be a multiple conical surface with generic plane section of genus one.
Then the linear normalization S of S is a del Pezzo surface.
Vice versa, let S be a surface such that its linear normalization S is a del Pezzo surface.
If S has several pencils of conics, then S is a multiple conical surface.
Proof. This is an immediate corollary of the properties of the linear normailzation described
in Section 2.
Here is a summary of the relevant facts on del Pezzo surfaces. For the proofs, we refer to
[11].
1. The degree of a del Pezzo surface is less than or equal to 9.
2. There is only one del Pezzo surface of degree 9, up to projective isomorphism. It is
embedded in P9 and has the parametrization
(1 : s : s2 : s3 : t : st : s2 t : t2 : st2 : t3 ).
The class group is generated by L : s = 0, and we have L2 = 1 and H∼3L.
3. There are three del Pezzo surfaces of degree 8, up to projective isomorphism. They are
embedded in P8 , and their parametrizations are
(1 : s : s2 : t : st : s2 t : t2 : st2 : s2 t2 ),
(1 : s : s2 : t : st : s2 t : t2 : st2 : t3 ),
(1 : t : t2 : t3 : t4 : s : st : st2 : s2 ).
The class group is generated by P : t = 0 and B : s = 0. We have P 2 = 0, P ·B = 1,
B 2 = −n, and H∼2B + (n + 2)P , where n = 0, 1, 2, respespectively.
4. For 3≤d≤7, any del Pezzo surface of degree d is embedded in Pd and has a parametrization by polynomials of total degree at most 3. The class group is a free abelian group of
rank 10−d, generated by divisors L, E1 , · · ·, E9−d . We have L2 = 1, Ei2 = −1, L·Ei = 0,
Ei ·Ej = 0 for i6=j, and H∼3L − E1 − . . . − E9−d .
5. The embedding divisor H is anticanonical (in fact, this property characterizes del Pezzo
surfaces).
6. A del Pezzo surface has only rational singularities. The preimage of the singularities
are unions of curves F such that F 2 = −2 and F ·K = 0, where K is the canonical
class (these curves are called -2-curves). Their intersection graphs are classified by the
Dynkin diagrams A1 , A2 , A3 , A4 , A5 , D4 , D5 , E6 .
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7. A del Pezzo surface has only finitely many lines. They correspond to curves E such
that E 2 = E·K = −1 (so-called -1-curves) on the desingularization.
All references to the class group refer to the minimal desingularization of S.
Remark 2. In the literature, one finds also del Pezzo surfaces of “degree” 1 and 2 (the
notion of degree is abused here). We do not need to consider these here.
Theorem 9. On a del Pezzo surfaces, any pencil of conics is the image of a pencil |P |, where
P is a divisor such that P 2 = 0 and P ·K = −2. All these divisors satisfy the conditions
P ·F ≥0, for all -2-curves F .
Proof. Let F be a pencil, and let P be the pullback of a generic conic in F. Then P ·K =
−P ·H = −2. By the genus formula, P 2 + P ·K = −2, hence P 2 = 0. Any curve in F must
be numerically equivalent to P , hence linearly equivalent, hence contained in |P |. Because
P 2 = 0, there is at most one curve in |P | passing through a fixed point. It follows that |P |
is a pencil and F = |P |.
Because P is an irreducible curve distinct from any -2-curve F , the intersection product
P ·F cannot be negative.
Theorem 10. A del Pezzo surface of degree 9 does not have any pencil of conics.
For d = 8, the pencils of conics are |P |, and |B| if n = 0.
For 3≤d≤7, the pencils of linear conics are among the linear families |L − Ei |, |2L − Ei1 −
Ei2 − Ei3 − Ei4 | where i1 , i2 , i2 , i4 are distinct (only for d≤5), and |3L − 2Ei1 − Ei2 − . . . − Ei6 |
where i1 , . . ., i6 are distinct (only for d = 3).
Proof. If d = 9, then P ·H is a multiple of 3 for each curve P , hence there are no conics at
all.
If d = 8, set Pi ∼xB + yP . By the conditions Pi2 = 0 and Pi ·H = 2, we get the equations
2xy − nx2 = 0,
−nx + 2y + 2x = 2,
with integral solutions (x, y) = (0, 1) and (x, y) = (1, n2 ) (only if n = 0 or n = 2). If n = 2,
then the second solution B + P violates the additional condition in Theorem 9, because
(B + P )·B = −1. For the remaining solutions, |xB + yP | is indeed a pencil of conics, as it
can easily be checked using the explicit parametrizations above.
If 3≤d≤7, then we set Pi ∼xL − x1 E1 − . . . − x9−d E9−d . The conditions Pi2 = 0 and
Pi ·H = 2 yield the equations
x2 = x21 + . . . + x29−d ,
3x − 2 = x1 + . . . + x9−d .
(1)
We assume that x1 ≥. . . ≥ x9−d (otherwise one has to permute the xi ). By the CauchySchwarz inequality, we have
(3x − 2)2 = (x1 + . . . + x9−d )2 ≤ (9 − d)(x21 + . . . + x29−d ) ≤ 6x2 .
The only integers satisfying (3x − 2)2 ≤6x2 are x = 1, 2, 3.
If x = 1, then the equations (1) give x1 = 1, x2 = . . . = x9−d = 0. If x = 2, then the
equations (1) give d≤5, x1 = x2 = x3 = x4 = 1, x5 = . . . = x9−d = 0. If x = 3, then the
equations (1) give d = 6, x1 = 2, x2 = . . . = x6 = 1.
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The theorem has the following converse.
Lemma 1. Let P be a divisor such that P 2 = 0, P ·H = 2, and P ·F ≥ 0 for each -2-curve
F . Then |P | is a pencil of conics on S.
Proof. Let R := P + H. We claim that R·C ≥ 0 for any curve C. Suppose, indirectly, that
R·C < 0 for an irreducible curve C. By the Riemann-Roch theorem,
dim|R|≥
R·(R + H)
− 1 = d − 2 ≥ 0,
2
hence C cannot move in a linear family. On a del Pezzo surface, this implies that C is a
-1-curve or a -2-curve (see [11]). Using the explicit description of the possible classes of -1curves in [17], one checks easily that P ·E≥0 for all these classes. Moreover, P ·F ≥0 for any
-2-curve F . This is a contradiction, and our claim is proven.
Now, we apply Reider’s theorem [20] and obtain that the linear family |P | does not
have base points. By Bertini’s theorem (see [12]), the generic element in |P | is nonsingular.
Because its arithmetic genus is zero, and its degree is 2, it must be an irreducible conic.
We continue the investigation by case distinction on the degree, following the classification
in [5].
Degree 8. The only multiple conical surface is the one with n = 0. The two pencils of
conics are formed by the curves s = constant and t = constant in the parametrization above.
Degree 7. If S is nonsingular, then we have a multiple conical surface with two pencils
|L − E1 |, |L − E2 |. If S is singular, then E1 − E2 is a -2-curve (maybe after renaming), and
then (L − E2 )·(E1 − E2 ) < 0, hence we have only one pencil.
Example 3. The surface S with parametrization
(x : y : z : w) = (s2 t + t + 1 : st2 + s + 1 : s2 + 1 : t2 + 1)
has an implicit equation of degree 7 (which is quite complicated). Its linear normalization
has the parametrization
(x : y : z : w : u1 : u2 : u3 : u4 ) =
= (s2 t + t + 1 : st2 + s + 1 : s2 + 1 : t2 + 1 : 1 : s : t : st).
This is a nonsingular del Pezzo surface of degree 7. The two pencils of conics are given
by s = constant and t = constant. Figure 4 shows the image of these two pencils on the
projection S.
Degree 6. Up to the choice of the orthogonal basis of the class group, the set D of -2-curves
is one of the following.
1. D = ∅ (S is nonsingular). Then we have three pencils of conics: |L − E1 |, |L − E2 |,
|L − E3 |.
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Figure 4: A surface of degree 7 with two pencils of conics
2. D = {E1 − E2 } (S has one double point of type A1 and 4 lines). Then we have two
pencils of conics: |L − E1 |, |L − E3 |. The divisor L − E2 fails to satisfy the additional
condition in Theorem 9.
3. D = {L − E1 − E2 − E3 } (S has one double point of type A1 and 3 lines). Then we
have three pencils of conics: |L − E1 |, |L − E2 |, |L − E3 |.
4. D = {L − E1 − E2 − E3 , E1 − E2 } (S has two double points of type A1 ). Then we have
two pencils of conics: |L − E1 |, |L − E3 |.
5. D = {E1 − E2 , E2 − E3 } (S has one double point of type A2 ). Then we have one pencil
of conics: |L − E1 |.
6. D = {L − E1 − E2 − E3 , E1 − E2 , E2 − E3 } (S has one double point of type A2 and one
double point of type A1 ). Then we have one pencil of conics: |L − E1 |.
Degree 5. Up to the choice of the orthogonal basis of the class group, the set D of -2-curves
is one of the following.
1. D = ∅ (S is nonsingular). Then we have 5 pencils of conics: |L − E1 |, |L − E2 |, |L − E3 |,
|L − E4 |, |2L − E1 − E2 − E3 − E4 |.
2. D = {E1 − E2 } (S has one double point of type A1 ). Then we have 4 pencils of conics:
|L − E1 |, |L − E3 |, |L − E4 |, |2L − E1 − E2 − E3 − E4 |.
3. D = {E1 − E2 , E3 − E4 } (S has two double points of type A1 ). Then we have 3 pencils
of conics: |L − E1 |, |L − E3 |, |2L − E1 − E2 − E3 − E4 |.
4. D = {E1 − E2 , E2 − E3 } (S has one double point of type A2 ). Then we have 3 pencils
of conics: |L − E1 |, |L − E4 |, |2L − E1 − E2 − E3 − E4 |.
5. D = {L − E1 − E2 − E3 , E1 − E2 , E2 − E3 } (S has one double point of type A2 and one
double point of type A1 ). Then we have 2 pencils of conics: |L − E1 |, |L − E4 |.
6. D = {E1 − E2 , E2 − E3 , E3 − E4 } (S has two double points of type A3 ). Then we have
2 pencils of conics: |L − E1 |, |2L − E1 − E2 − E3 − E4 |.
7. D = {L − E1 − E2 − E3 , E1 − E2 , E2 − E3 , E3 − E4 } (S has two double points of type
A4 ). Then we have one pencil of conics: |L − E1 |.
J. Schicho: The Multiple Conical Surfaces
83
Degree 4. Here, one can observe the phenomenon of complementary pencils: if a hyperplane
contains a conic, then it contains also a second one because the degree of any hyperplane
section is 4.
Lemma 2. The pencil complementary to |P | can be constructed by the following procedure:
Initialize P 0 := H − P ; while there exists a -2-curve F such that F ·P 0 < 0, replace P 0 by
P0 − F.
Proof. Any curve that has negative intersection number with H − P must be a fixed component of |H − P |. Because the number of fixed components is finite, the procedure above
terminates.
If e := −F ·(H − P ) > 1, then P ·(2F + eH) = 0 and (2F + eH)2 > 0. By the Hodge index
theorem, it follows that P is numerically zero, which is a contradiction. Hence F ·(H −P ) < 0
implies F ·(H − P ) = −1. Thus, (H − P − F )2 = 0. By induction over the while loop, we
see that (P 0 )2 = 0. By Lemma 1, |P 0 | is a pencil of lines. Moreover, there is a hyperplane
containing P + P 0 , hence |P 0 | is complementary to |P |.
Using the tables of possible sets of -2-curves (up to choice of the orthogonal basis) in [5]
and the theorems 9 and 10, we can compute all possible configurations of pencils of conics.
The result is displayed in Figure 4. Complementary pencils are indicated by a dash between
them. The pencils in the table without attaching dash are self-complementary.
Example 4. The torus, with homogeneous equation
(x2 + y 2 + z 2 + (1 − r2 )w2 )2 − 4x2 − 4y 2 = 0,
where r is a parameter in the open interval (0, 1), has the linear normalization
x2 + y 2 + z 2 − wu = (u + (1 − r2 )w)2 − 4x2 − 4y 2 = 0
in P4 . This is a del Pezzo surface with four double points of type A1 , namely (1 : ±i : 0 :
0 : 0), (0 : 0 : ±i(1 − r) : 1 : −(1 − r)2 ). According to Figure 5, the surface has four pencils
of conics, two of them being self-complementary. The two pencils are shown in Figure 6.
Indeed, the two formed by the rotating circle and by the orbits under this rotation are selfcomplementary. The pair of complementary pencils is cut out by the tangent planes through
the origin.
Degree 3. (Cubic surfaces with at most isolated double points) For any line on S, the
planes through the lines cut out a pencil of conics. Vice versa, let P be an irreducible conic.
Then the plane carrying P intersects S in P and a line L, and P is contained in the pencil
corresponding to L. Because any irreducible conic is contained in a unique pencil (namely
|P |), we have a one-to-one correspondence between pencils of conics and lines.
According to the classification in [3], the number of lines can be 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 15, 16, 21, or 27. Thus, we have a multiple conical surface in all these cases except
the first.
84
J. Schicho: The Multiple Conical Surfaces
-2-curves
none
singularities
none
E4 − E5
1 A1
L − E1 − E2 − E3 ,
E4 − E5
2 A1
E3 − E4 , E 4 − E5
1 A2
L − E1 − E2 − E3 ,
E2 − E3 , E 4 − E5
3 A1
E1 − E2 , E 3 − E4 ,
E4 − E5
E2 − E3 , E 3 − E4 ,
E4 − E5
L − E1 − E2 − E3 ,
E3 − E4 , E 4 − E5
1 A1 , 1 A2
L − E1 − E2 − E3 ,
L − E3 − E4 − E5 ,
E1 − E2 , E 4 − E5
L − E1 − E2 − E3 ,
E1 − E2 , E 2 − E3 ,
E4 − E5
L − E1 − E2 − E3 ,
E1 − E2 , E 3 − E4 ,
E4 − E5
E1 − E2 , E 2 − E3 ,
E3 − E4 , E 4 − E5
L − E1 − E2 − E3 ,
E2 − E3 , E 3 − E4 ,
E4 − E5
L − E1 − E2 − E3 ,
L − E3 − E4 − E5 ,
E1 − E2 , E 3 − E4 ,
E4 − E5
L − E1 − E2 − E3 ,
E1 − E2 , E 2 − E3 ,
E3 − E4 , E 4 − E5
4 A1
1 A3
1 A3
pencils of conics
|L − E1 | - |2L − E2 − E3 − E4 − E5 |
...
|L − E5 | - |2L − E1 − E2 − E3 − E4 |
|L − E1 | - |2L − E2 − E3 − E4 − E5 |
|L − E2 | - |2L − E1 − E3 − E4 − E5 |
|L − E3 | - |2L − E1 − E2 − E4 − E5 |
|L − E4 | - |2L − E1 − E2 − E3 − E4 |
|L − E1 | - |2L − E2 − E3 − E4 − E5 |,
|L − E2 | - |2L − E1 − E3 − E4 − E5 |,
|L − E3 | - |2L − E1 − E2 − E4 − E5 |,
|L − E4 |
|L − E1 | - |2L − E2 − E3 − E4 − E5 |,
|L − E2 | - |2L − E1 − E3 − E4 − E5 |,
|L − E3 | - |2L − E1 − E2 − E3 − E4 |
|L − E1 | - |2L − E2 − E3 − E4 − E5 |,
|L − E2 | - |2L − E1 − E2 − E4 − E5 |,
|L − E4 |
|L − E1 | - |2L − E1 − E3 − E4 − E5 |,
|L − E3 | - |2L − E1 − E2 − E3 − E4 |
|L − E1 | - |2L − E2 − E3 − E4 − E5 |,
|L − E2 | - |2L − E1 − E2 − E3 − E4 |
|L − E1 | - |2L − E2 − E3 − E4 − E5 |,
|L − E2 | - |2L − E1 − E3 − E4 − E5 |,
|L − E3 |
|L − E3 | - |2L − E1 − E2 − E4 − E5 |,
|L − E1 |, |L − E4 |
2 A1 , 1 A2
|L − E1 | - |2L − E1 − E2 − E4 − E5 |,
|L − E4 |
1 A1 , 1 A3
|L − E1 | - |2L − E1 − E3 − E4 − E5 |,
|L − E3 |
1 A4
|L − E1 | - |2L − E1 − E2 − E3 − E4 |
1 D4
|L − E1 | - |2L − E2 − E3 − E4 − E5 |,
|L − E2 |
2 A1 , 1 A3
|L − E1 |, |L − E3 |
1 D5
|L − E1 |
Figure 5: Del Pezzo Surfaces of degree 4
J. Schicho: The Multiple Conical Surfaces
85
Figure 6: The torus and its four pencils of conics
Example 5. The cubic surface with equation
z(x2 + y 2 + z 2 ) − xyw = 0
(see also Figure 1) has a double point of type A2 at the point (0 : 0 : 0 : 1). According to
the classification in [3], it has 15 lines. Twelve of them are complex. The three real lines
are x = z = 0, y = z = 0, and w = z = 0. The pencils of conics corresponding to the
two lines through the singular point consist of circles, and they are shown in figure 1. The
pencil corresponding to the infinite line w = z = 0 is the pencil of horizontal sections. From
the implicit equation (considering z as a constant), it is clear that we have hyperbolas for
z∈(−1/2, 1/2), and empty sections outside this interval.
The following algebraic characterization of multiple conical surfaces can be proved through
the classification above.
Theorem 11. Any multiple conical surface has a parametrization
(x : y : z : w) = (X(s, t) : Y (s, t) : Z(s, t) : W (s, t)),
where the maximum of the degrees in s and in t of X, Y, Z, W is 2.
Proof. For the quadric surface, the claim is obviously true. For the ruled surface R1,1 and for
the Veronese surface, we have already mentioned parametrizations of the desired type. The
same holds for del Pezzo surfaces of degree 8 (recall that we must have n = 0 in this case).
It remains to show the statement for del Pezzo surfaces of degree less than or equal to 7.
Explicit checking in the classification above shows that there exist always two pencils
|P1 | and |P2 | with P1 ·P2 = 1. Their generic conics intersect in a single point. Hence, the
product of the two structure maps i1 ×i2 : S→P1 ×P1 is birational. The inverse P1 ×P1 →S
is a parametrization as desired. Indeed, the two pencils are s = constant and t = constant;
because these curves are conics, the degree of the parametrization must be equal 2 in both
variables.
Remark 3. The converse of Theorem 11 does not hold, because the ruled surface R0,2 has
a degree 2 parametrization but only one pencil of conics.
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J. Schicho: The Multiple Conical Surfaces
Remark 4. Over the reals, Theorem 11 does not hold, as one can construct del Pezzo
surfaces which do not have two real pencils with intersection product equal to one. An
example is the cubic surface yx2 + w3 + wz 2 = 0. It has a D4 singularity at (0 : 1 : 0 : 0).
According to the classification in [3], there are 6 lines. These lines are given by x = w = 0,
y = w = 0, x = w±iz = 0, y = w±iz = 0. Only the first two lines are real and correspond
to real pencils. These two pencils have intersection number 2; indeed the generic elements
given by x = sw and y = tw intersect in two varying points
√
(x : y : z : w) = (s : t : −s2 t − 1 : 1).
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Received November 5, 1999